Per the SCSW (Standard Catalog of Smith & Wesson) 4th edition the 34-1 was introduced in 1960 at about serial # 70000 and the 34-1 was produced until about 1988 when the 34-2 was introduced. Thumbpiece was changed to more modern style which your revolver has in 1966. M prefix serial number was introduced in 1969. Pinned barrel eliminated in 1982. Just by looking at your revolver it would date to between 1966 (after new thumbpiece) and 1982 (pinned barrel deleted) however the serial number you provide (with no M prefix) would date your revolver to somewhere between about 1956 and 1959.
My guess is that you left the M prefix off your serial number? If so that would date your revolver to 1981. The best place to find the serial number is on the bottom of the butt frame (under the grips on your revolver).